Measure entanglement with one matrix decomposition

Write the amplitudes of a chain as a matrix whose rows are the patterns of the left part and whose columns are the patterns of the right part. A singular value decomposition of that matrix rewrites the state as a short sum of left states times right states. The squared singular values, the Schmidt weights, add up to one, and their entropy is the entanglement across the cut.

Sandbox Physics · 2026-10-29

Reshape the list into a matrix

Cut a chain of twelve spins after the fourth spin. The left part has sixteen patterns and the right part has 256. Put the amplitude of each full pattern in the row of its left part and the column of its right part. The state is now a sixteen by 256 matrix, and nothing has been lost.

If the matrix has rank one, every row is a multiple of the same row. The state is then a single left state times a single right state, a product state, and the two sides are independent. Higher rank means the two sides are entangled: what the right part looks like depends on which left pattern you find.

\Psi_{(s_1\cdots s_\ell),(s_{\ell+1}\cdots s_L)}=\psi_{s_1\cdots s_L},\qquad \Psi\in\mathbb C^{2^{\ell}\times2^{L-\ell}}

Singular values become Schmidt weights

The singular value decomposition writes the matrix as a sum of rank-one pieces, each a column of left vectors times a row of right vectors, scaled by a non-negative singular value. The left vectors are orthonormal, and so are the right vectors. Read back as quantum states, this is the Schmidt decomposition. Schollwöck sets it out in Section 4.1.1 of his review.

For a normalized state the squared singular values add up to one, so they behave like probabilities. They are the Schmidt weights. The entanglement entropy is the entropy of these weights. It is zero for one weight equal to one, and it is largest, the logarithm of the number of terms, when all the weights are equal.

|\psi\rangle=\sum_{\alpha=1}^{r}s_\alpha\,|a_\alpha\rangle_{\rm L}|b_\alpha\rangle_{\rm R},\qquad \lambda_\alpha=s_\alpha^2,\qquad S=-\sum_{\alpha}\lambda_\alpha\ln\lambda_\alpha\le\ln r

Worked example

Work out two spins and a cat state

Calabrese and Cardy open their 2004 paper with two spins in a state that has the first spin up and the second down with one amplitude, and the reverse with another. As a matrix with the first spin as rows and the second as columns, the state has only off-diagonal entries. Its singular values are the magnitudes of the two amplitudes, so the Schmidt weights are their squares. When both weights are one half, the entropy reaches its largest value for two spins, the natural logarithm of two.

Now take a chain whose spins all point the same way, in a superposition of all along one direction and all along the opposite direction, with equal weight. Every cut gives exactly two Schmidt weights of one half, so every cut carries the logarithm of two, about 0.693. The compression Lab cat-state preset is the Ising ground state at a field of 0.3, close to this picture. Its middle cut carries 0.6935.

\cos\theta\,|{\uparrow\downarrow}\rangle+\sin\theta\,|{\downarrow\uparrow}\rangle\ \Rightarrow\ \Psi=\begin{pmatrix}0&\cos\theta\\ \sin\theta&0\end{pmatrix},\quad S=-\cos^2\theta\ln\cos^2\theta-\sin^2\theta\ln\sin^2\theta

The entropy along a critical chain

Move the cut along the chain and record the entropy at each position. For a gapped chain the curve quickly flattens to a constant away from the ends. For a critical chain it keeps rising towards the middle. Calabrese and Cardy derived its shape for an open critical chain from conformal field theory, in their Eq. 2. The coefficient is one sixth of the central charge, which is one half for the critical Ising chain.

You can test that shape without any approximation. Peschel showed that for chains that map to free fermions, including those with pairing terms such as the Ising chain, the Schmidt weights of a block follow from its correlation matrix. Fitting the profile of the exact ground state over the middle half of the chain gives a central charge of 0.524 for 32 spins, then 0.512, 0.506 and 0.503 for 64, 128 and 256 spins. The fit approaches one half as the chain grows. The DMRG Lab plots the same law, with the central charge fixed and only the constant fitted, beside the profile its run measures.

S(\ell)=\frac{c}{6}\ln\!\Big[\frac{L}{\pi}\sin\frac{\pi\ell}{L}\Big]+s_0,\qquad c_{\rm Ising}=\tfrac12

Predict, change, explain

In the DMRG Lab, run the critical Ising chain and look at the entanglement bars. Predict where they are tallest and how fast they fall towards the ends.

Switch to the gapped Ising chain.

Explain why the bars of the gapped chain flatten away from the ends while those of the critical chain keep rising towards the middle.

The Labs simulate open spin-1/2 chains with energies in units of the coupling and time in units of the reduced Planck constant divided by the coupling. Each Lab states its model and limits in its notes and can export the data behind its readouts.

Try explaining it yourself

1. The state with every spin pointing up along the field axis is cut in the middle. What are its Schmidt weights and its entropy?

Read an explained answer

It is a product state, so its matrix has rank one. There is a single Schmidt weight equal to one and the entropy is zero.

2. A cut keeps at most four Schmidt weights. What is the largest entropy that cut can carry?

Read an explained answer

The entropy is largest when the four weights are equal, each one quarter. It is then the natural logarithm of four, about 1.39.

S\le\ln4\approx1.386

3. Why does the fitted central charge come out at 0.524 for 32 spins, and where does it go as the chain grows?

Read an explained answer

The formula is the leading behaviour of a long chain. A short chain adds corrections that fall off with its length, so the fit drifts towards one half as the chain grows: 0.512 at 64 spins and 0.503 at 256.

Sources to read alongside

U. Schollwöck, The density-matrix renormalization group in the age of matrix product states, Annals of Physics 326, 96 (2011)

The standard review. Section 3 uses the area law to explain why DMRG works in one dimension; Section 4.1 builds matrix product states from successive singular value decompositions and bounds the truncation error in Eq. 58; Section 6 sets out the ground-state search as a variational sweep; Section 7 covers time evolution and its limits.

P. Calabrese and J. Cardy, Entanglement entropy and quantum field theory, Journal of Statistical Mechanics P06002 (2004)

Derives the entanglement entropy of critical one-dimensional systems. Equation 2 gives the profile along an open chain; its coefficient is one sixth of the central charge.

I. Peschel, Calculation of reduced density matrices from correlation functions, Journal of Physics A 36, L205 (2003)

Shows that for free-fermion chains the reduced density matrix of a block follows from its two-point correlation matrix. The exact entanglement used as a reference in the Labs comes from this route.